[SOLVED] The Bear and the Food

Icebreaker

5 friends go camping, and bring with them a number of packets of food. In the middle of the night, one of them wakes up, sees a hungry bear, and decides to give the bear one packet of food. He then proceeds to divide the remaining packets of food into 5 equal parts, and takes one part for himself, then goes back to bed. A second friend wakes up, sees the bear, gives the bear a packet of food, then proceeds to, as with the first, divide the remaining food into 5 parts and keeps one for himself. The third, fourth and fifth each wakes up one after another and does the same thing. What is the minimum packets of food in the beginning.

There are 2 possible answers that I'm aware of.

BicycleTree

There are a number of possible answers that I'm aware of:

3121, 6246, 9371, 12496, 15621, 18746, 21871, 24996, 28121, 31246, 34371, 37496, 40621, 43746, 46871, 49996, 53121, 56246, 59371, 62496, 65621, 68746, 71871, 74996, 78121, 81246, 84371, 87496, 90621, 93746, 96871, 99996, 103121, 106246, 109371, 112496, 115621, 118746, 121871, 124996, 128121, 131246, 134371, 137496, 140621, 143746, 146871, 149996, 153121, 156246, 159371, 162496, 165621, 168746, 171871, 174996, 178121, 181246, 184371, 187496, 190621, 193746, 196871, 199996, 203121, 206246, 209371, 212496, 215621, 218746, 221871, 224996, 228121, 231246, 234371, 237496, 240621, 243746, 246871, 249996, 253121, 256246, 259371, 262496, 265621, 268746, 271871, 274996, 278121, 281246, 284371, 287496, 290621, 293746, 296871, 299996, 303121, 306246, 309371, 312496, 315621, 318746, 321871, 324996, 328121, 331246, 334371, 337496, 340621, 343746, 346871, 349996, 353121, 356246, 359371, 362496, 365621, 368746, 371871, 374996, 378121, 381246, 384371, 387496, 390621, 393746, 396871, 399996, 403121, 406246, 409371, 412496, 415621, 418746, 421871, 424996, 428121, 431246, 434371, 437496, 440621, 443746, 446871, 449996, 453121, 456246, 459371, 462496, 465621, 468746, 471871, 474996, 478121, 481246, 484371, 487496, 490621, 493746, 496871, 499996, 503121, 506246, 509371, 512496, 515621, 518746, 521871, 524996, 528121, 531246, 534371, 537496, 540621, 543746, 546871, 549996, 553121, 556246, 559371, 562496, 565621, 568746, 571871, 574996, 578121, 581246, 584371, 587496, 590621, 593746, 596871, 599996, 603121, 606246, 609371, 612496, 615621, 618746, 621871, 624996, 628121, 631246, 634371, 637496, 640621, 643746, 646871, 649996, 653121, 656246, 659371, 662496, 665621, 668746, 671871, 674996, 678121, 681246, 684371, 687496, 690621, 693746, 696871, 699996, 703121, 706246, 709371, 712496, 715621, 718746, 721871, 724996, 728121, 731246, 734371, 737496, 740621, 743746, 746871, 749996, 753121, 756246, 759371, 762496, 765621, 768746, 771871, 774996, 778121, 781246, 784371, 787496, 790621, 793746, 796871, 799996, 803121, 806246, 809371, 812496, 815621, 818746, 821871, 824996, 828121, 831246, 834371, 837496, 840621, 843746, 846871, 849996, 853121, 856246, 859371, 862496, 865621, 868746, 871871, 874996, 878121, 881246, 884371, 887496, 890621, 893746, 896871, 899996, 903121, 906246, 909371, 912496, 915621, 918746, 921871, 924996, 928121, 931246, 934371, 937496, 940621, 943746, 946871, 949996, 953121, 956246, 959371, 962496, 965621, 968746, 971871, 974996, 978121, 981246, 984371, 987496, 990621, 993746, 996871, 999996, 1003121, 1006246, 1009371, 1012496, 1015621, 1018746, 1021871, 1024996, 1028121, 1031246, 1034371, 1037496, 1040621, 1043746, 1046871, 1049996, 1053121, 1056246, 1059371, 1062496, 1065621, 1068746, 1071871, 1074996, 1078121, 1081246, 1084371, 1087496, 1090621, 1093746, 1096871, 1099996, 1103121, 1106246, 1109371, 1112496, 1115621, 1118746, 1121871, 1124996, 1128121, 1131246, 1134371, 1137496, 1140621, 1143746, 1146871, 1149996, 1153121, 1156246, 1159371, 1162496, 1165621, 1168746, 1171871, 1174996, 1178121, 1181246, 1184371, 1187496, 1190621, 1193746, 1196871, 1199996, 1203121, 1206246, 1209371, 1212496, 1215621, 1218746, 1221871, 1224996, 1228121, 1231246, 1234371, 1237496, 1240621, 1243746, 1246871, 1249996, 1253121, 1256246, 1259371, 1262496, 1265621, 1268746, 1271871, 1274996, 1278121, 1281246, 1284371, 1287496, 1290621, 1293746, 1296871, 1299996, 1303121, 1306246, 1309371, 1312496, 1315621, 1318746, 1321871, 1324996, 1328121, 1331246, 1334371, 1337496, 1340621, 1343746, 1346871, 1349996, 1353121, 1356246, 1359371, 1362496, 1365621, 1368746, 1371871, 1374996, 1378121, 1381246, 1384371, 1387496, 1390621, 1393746, 1396871, 1399996, 1403121, 1406246, 1409371, 1412496, 1415621, 1418746, 1421871, 1424996, 1428121, 1431246, 1434371, 1437496, 1440621, 1443746, 1446871, 1449996, 1453121, 1456246, 1459371, 1462496, 1465621, 1468746, 1471871, 1474996, 1478121, 1481246, 1484371, 1487496, 1490621, 1493746, 1496871, 1499996, 1503121, 1506246, 1509371, 1512496, 1515621, 1518746, 1521871, 1524996, 1528121, 1531246, 1534371, 1537496, 1540621, 1543746, 1546871, 1549996, 1553121, 1556246, 1559371, 1562496, 1565621, 1568746, 1571871, 1574996, 1578121, 1581246, 1584371, 1587496, 1590621, 1593746, 1596871, 1599996, 1603121, 1606246, 1609371, 1612496, 1615621, 1618746, 1621871, 1624996, 1628121, 1631246, 1634371, 1637496, 1640621, 1643746, 1646871, 1649996, 1653121, 1656246, 1659371, 1662496, 1665621, 1668746, 1671871, 1674996, 1678121, 1681246, 1684371, 1687496, 1690621, 1693746, 1696871, 1699996, 1703121, 1706246, 1709371, 1712496, 1715621, 1718746, 1721871, 1724996, 1728121, 1731246, 1734371, 1737496, 1740621, 1743746, 1746871, 1749996, 1753121, 1756246, 1759371, 1762496, 1765621, 1768746, 1771871, 1774996, 1778121, 1781246, 1784371, 1787496, 1790621, 1793746, 1796871, 1799996, 1803121, 1806246, 1809371, 1812496, 1815621, 1818746, 1821871, 1824996, 1828121, 1831246, 1834371, 1837496, 1840621, 1843746, 1846871, 1849996, 1853121, 1856246, 1859371, 1862496, 1865621, 1868746, 1871871, 1874996, 1878121, 1881246, 1884371, 1887496, 1890621, 1893746, 1896871, 1899996, 1903121, 1906246, 1909371, 1912496, 1915621, 1918746, 1921871, 1924996, 1928121, 1931246, 1934371, 1937496, 1940621, 1943746, 1946871, 1949996, 1953121, 1956246, 1959371, 1962496, 1965621, 1968746, 1971871, 1974996, 1978121, 1981246, 1984371, 1987496, 1990621, 1993746, 1996871, 1999996, 2003121, 2006246, 2009371, 2012496, 2015621, 2018746, 2021871, 2024996, 2028121, 2031246, 2034371, 2037496, 2040621, 2043746, 2046871, 2049996, 2053121, 2056246, 2059371, 2062496, 2065621, 2068746, 2071871, 2074996, 2078121, 2081246, 2084371, 2087496, 2090621, 2093746, 2096871, 2099996, 2103121, 2106246, 2109371, 2112496, 2115621, 2118746, 2121871, 2124996, 2128121, 2131246, 2134371, 2137496, 2140621, 2143746, 2146871, 2149996, 2153121, 2156246, 2159371, 2162496, 2165621, 2168746, 2171871, 2174996, 2178121, 2181246, 2184371, 2187496, 2190621, 2193746, 2196871, 2199996, 2203121, 2206246, 2209371, 2212496, 2215621, 2218746, 2221871, 2224996, 2228121, 2231246, 2234371, 2237496, 2240621, 2243746, 2246871, 2249996, 2253121, 2256246, 2259371, 2262496, 2265621, 2268746, 2271871, 2274996, 2278121, 2281246, 2284371, 2287496, 2290621, 2293746, 2296871, 2299996, 2303121, 2306246, 2309371, 2312496, 2315621, 2318746, 2321871, 2324996, 2328121, 2331246, 2334371, 2337496, 2340621, 2343746, 2346871, 2349996, 2353121, 2356246, 2359371, 2362496, 2365621, 2368746, 2371871, 2374996, 2378121, 2381246, 2384371, 2387496, 2390621, 2393746, 2396871, 2399996, 2403121, 2406246, 2409371, 2412496, 2415621, 2418746, 2421871, 2424996, 2428121, 2431246, 2434371, 2437496, 2440621, 2443746, 2446871, 2449996, 2453121, 2456246, 2459371, 2462496, 2465621, 2468746, 2471871, 2474996, 2478121, 2481246, 2484371, 2487496, 2490621, 2493746, 2496871, 2499996, 2503121, 2506246, 2509371, 2512496, 2515621, 2518746, 2521871, 2524996, 2528121, 2531246, 2534371, 2537496, 2540621, 2543746, 2546871, 2549996, 2553121, 2556246, 2559371, 2562496, 2565621, 2568746, 2571871, 2574996, 2578121, 2581246, 2584371, 2587496, 2590621, 2593746, 2596871, 2599996, 2603121, 2606246, 2609371, 2612496, 2615621, 2618746, 2621871, 2624996, 2628121, 2631246, 2634371, 2637496, 2640621, 2643746, 2646871, 2649996, 2653121, 2656246, 2659371, 2662496, 2665621, 2668746, 2671871, 2674996, 2678121, 2681246, 2684371, 2687496, 2690621, 2693746, 2696871, 2699996, 2703121, 2706246, 2709371, 2712496, 2715621, 2718746, 2721871, 2724996, 2728121, 2731246, 2734371, 2737496, 2740621, 2743746, 2746871, 2749996, 2753121, 2756246, 2759371, 2762496, 2765621, 2768746, 2771871, 2774996, 2778121, 2781246, 2784371, 2787496, 2790621, 2793746, 2796871, 2799996, 2803121, 2806246, 2809371, 2812496, 2815621, 2818746, 2821871, 2824996, 2828121, 2831246, 2834371, 2837496, 2840621, 2843746, 2846871, 2849996, 2853121, 2856246, 2859371, 2862496, 2865621, 2868746, 2871871, 2874996, 2878121, 2881246, 2884371, 2887496, 2890621, 2893746, 2896871, 2899996, 2903121, 2906246, 2909371, 2912496, 2915621, 2918746, 2921871, 2924996, 2928121, 2931246, 2934371, 2937496, 2940621, 2943746, 2946871, 2949996, 2953121, 2956246, 2959371, 2962496, 2965621, 2968746, 2971871, 2974996, 2978121, 2981246, 2984371, 2987496, 2990621, 2993746, 2996871, 2999996, 3003121, 3006246, 3009371, 3012496, 3015621, 3018746, 3021871, 3024996, 3028121, 3031246, 3034371, 3037496, 3040621, 3043746, 3046871, 3049996.

BicycleTree

Although no camper is likely to bring THAT much food, unless the packets are very very small.

Icebreaker

I forgot the word "minimum".

Btw, I randomly tested a few of your answers, and they were all wrong.

BicycleTree

They all seem to be of the form 3121 + 3125n, though I couldn't tell you why.

BicycleTree

Quote by Icebreaker

I forgot the word "minimum".

Btw, I randomly tested a few of your answers, and they were all wrong.

Seriously? Hold on let me check.

Icebreaker

Quote by BicycleTree

Although no camper is likely to bring THAT much food, unless the packets are very very small.

The number does not matter as long as it fits the conditions. Think of it as a math problem in disguise.

BicycleTree

Well, I'm thinking that the "packets" are peanut or cashew shells.

I checked 3121 and it does work.
Beginning:
3121 cashews
After camper 1:
(3121 - 1)*4/5 = 2496
After camper 2:
(2496 - 1)*4/5 = 1996
After camper 3:
(1996 - 1)*4/5 = 1596
After camper 4:
(1596 - 1)*4/5 = 1276
After camper 5:
(1276 - 1)*4/5 = 1020

medshredr

If the contents of each individual packet cannot be divided and resealed, at some point the remaining packets will not be divisible by 5 and this teaser does not work. Assuming the individual contents can be divided and resealed, any number other than 1 will work.

BicycleTree

By the way, I solved this by brute force (wrote a program). I'm interested to know how you would do it algebraically.

Icebreaker

Quote by medshredr

If the contents of each individual packet cannot be divided and resealed, at some point the remaining packets will not be divisible by 5 and this teaser does not work. Assuming the individual contents can be divided and resealed, any number other than 1 will work.

Well I used the word "packets" hoping that it would invoke the word "quanta". Anyway, the packets cannot be subdivided.

I tested the first few answers, and they seem to be right.

I'll post an algebraic solution later. But even with the word "minimum", there are still 2 possible answers. One is more "abstract" than the other.

BicycleTree

This is my program:

Code:

class BearFood{
	public static void main(String arg[])
	{
		int n;
		boolean stillok=true;
		for(int i = 1; i < 1000000; i++)
		{
			n = i;
			stillok=true;
			for(int j = 1; j <=5; j++)
			{
				if(n % 4 == 0)
					n = n*5/4 + 1;
				else{
					stillok=false;
					break;
				}
			}
			if(stillok==true)
				System.out.print(n + ", ");
		}
	}
}

Icebreaker

[tex]m = \frac{4}{5}n-\sum_{i = 0}^{5}\frac{4^i}{5^i}[/tex], where [tex]\{m, n\}[/tex] are integers.

The graph will show all possible solutions.

The other "elegant" solution is -4.

ArielGenesis

bicycletree, you seems to be familiar with c ++

quark

Did the lengthy algebraic expression. If quotient of (256x-2101)/625 when divided by 5 is a natural number, you will get the answer. Above given numbers fit well.

BicycleTree

Quote by ArielGenesis

bicycletree, you seems to be familiar with c ++

Well, I have had a course in C++ but right now I use Java.

BicycleTree

Quote by quark

Did the lengthy algebraic expression. If quotient of (256x-2101)/625 when divided by 5 is a natural number, you will get the answer. Above given numbers fit well.

They all are of the form 3121 (mod 3125). Now where does the 3121 come from?

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BicycleTree	Although no camper is likely to bring THAT much food, unless the packets are very very small.

Icebreaker	[SOLVED] The Bear and the Food I forgot the word "minimum". Btw, I randomly tested a few of your answers, and they were all wrong.

medshredr	If the contents of each individual packet cannot be divided and resealed, at some point the remaining packets will not be divisible by 5 and this teaser does not work. Assuming the individual contents can be divided and resealed, any number other than 1 will work.

quark	Did the lengthy algebraic expression. If quotient of (256x-2101)/625 when divided by 5 is a natural number, you will get the answer. Above given numbers fit well.